This relates to apparatus for parallel processing of signals.
In spite of the tremendous computational power of digital computers, for a variety of reasons many practical problems cannot effectively be solved by the digital computer. The reason may be that a closed form solution is unavailable for the particular formulation of the problem and, therefore, numerical methods must be employed; or it may be that the number of variables in an optimization problem is large, and finding the optimum solution from the very large set of possible solutions is just too great a task to be performed by a digital computer in a reasonable time.
Heretofore, most artisans have either suffered the limitations of general purpose digital computers, or developed special purpose digital computers to solve their particular problems more efficiently. Recently, advances have brought to the forefront a class of highly parallel computation circuits that solve a large class of complex problems in analog fashion. These circuits comprise a plurality of amplifiers having a sigmoid transfer function and a resistive feedback network that connects the output of each amplifier to the input of the other amplifiers. Each amplifier input also includes a capacitor connected to ground and a resistor connected to ground. Input currents are fed into each amplifier input, and output is obtained from the collection of output voltages of the amplifiers.
A generalized diagram of this circuit is shown in FIG. 1, depicting five amplifiers (10-14) with positive and negative outputs V.sub.1 and -V.sub.1 on line pair 21, V.sub.2 and -V.sub.2 on line pair 22, V.sub.3 and -V.sub.3 on line pair 23, V.sub.4 and -V.sub.4 on line pair 24, and V.sub.N and -V.sub.N on line pair 25. Those outputs are connected to an interconnection block 20 which has output lines 40-44 connected to the input ports of amplifiers 10-14, respectively. Within interconnection block 20, each output voltage V.sub.i is connected to each and every output line of block 20 through a conductance (e.g., resistor). For convenience, the conductance may be identified by the specific output line (i.e., source) that is connected by the conductance to a specific voltage line. For example, T.sub.21.sup.+ identifies the conductance that connects the positive output V.sub.2 of amplifier 11 to the input of the first amplifier (line 41).
Also connected to each amplifier input port is a resistor and a capacitor whose second lead is grounded, and means for injecting a current (from some outside source) into each input port.
Applying Kirchoff's current law to the input port of each amplifier i of FIG. 1 yields the equation: ##EQU1## where C.sub.i is the capacitance between the input of amplifier i and ground,
1/R.sub.i is the equivalent resistance and it equals ##EQU2## where .rho..sub.i is the resistance between the input of amplifier i and ground, PA0 u.sub.i is the voltage at the input of amplifier i, PA0 T.sub.ij.sup.+ is the a conductance between the positive output of amplifier j and the input of amplifier i, PA0 T.sub.ij.sup.- is the a conductance between the negative output of amplifier j and the input of amplifier i, PA0 V.sub.j is the positive output voltage of amplifier j, and PA0 I.sub.i is the current driven into the input port of amplifier i by an external source.
When T.sub.ij.sup.+ and T.sub.ij.sup.- are disjoint, T.sub.ij.sup.+ -T.sub.ij.sup.- may for convenience be expressed as T.sub.ij, and it is well known that a circuit satisfying Equation (1) with symmetric T.sub.ij terms is stable. It is also well known that such a circuit responds to applied stimuli, and reaches a steady state condition after a short transition time. At steady state, du.sub.i /dt=0 and dV.sub.i /dt=0.
With this known stability in mind, the behavior of other functions may be studied which relate to the circuit of FIG. 1 and involve the input signals of the circuit, the output signals of the circuit, and/or the circuit's internal parameters.
Indeed, in a copending application entitled Optimization Network for the Decompositon of Signals, by J. J. Hopfield, a function was studied that has the form ##EQU3## It is observed in this copending application that the integral of the function g.sub.i.sup.-1 (V) approaches 0 as the gain of amplifier i approaches infinity. It is also shown in the Hopfield application that the time derivative of the function E is negative, and that it reaches 0 when the time derivative of voltages V.sub.i reaches 0. Since equation (1) assures the condition of dV.sub.i /dt approaching 0 for all i, the function E of equation (2) is assured of reaching a stable state. The discovery of this function E led to the use of the FIG. 1 circuit in problem solving applications (such as the classic traveling salesman problem), in associative memory applications, and in decomposition problems (as disclosed in another copending application, by J. J. Hopfield and D. W. Tank, titled Optimization Network for the Decomposition of Signals).
The FIG. 1 circuit can solve the above classes of problems when those problems are structured so that the minimization of a function having at most second order terms in some parameters of the problem resuls in the desired solution. Other problems, however, may require the minimization of equations that contain terms of order higher than two. Those may be problems that perhaps can otherwise be stated with second order terms, but the statement with higher order terms is more meaningful, or they may be problems that can only be described with the use of higher order terms.
It is the object of this invention, therefore, to obtain solutions for problems characterized by higher order terms with a highly parallel circuit not unlike the circuit of FIG. 1.